C₂₉×#1517 #16815231

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#1517

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31)(32,36)(33,35)(34,37)(38,39)(40,46,48,43)(41,44)(42,45)(47,52,49,51)(50,53);(31,35)(32,33)(36,37)(38,40)(39,43)(41,49)(42,48)(44,52)(45,46)(47,50)(51,53);(30,32,34,33)(31,35,37,36)(38,41,42,50)(39,44,45,53)(40,47,48,49)(43,51,46,52)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8428, C₂₉×#8429, C₂₉×#8430, …
maximal normal subgroup of	C₂₉×#8428, C₂₉×#8429, C₂₉×#8430, …
maximal quotient of	C₂₉×#7574, C₂₉×#7587, C₂₉×#7588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 249);